23H (source code)
= 23H
{scope}
{title2=Analysis of Functions}
The <Schwartz space> $\mathcal S(\mathbb R^n)$ consists of smooth functions for which every seminorm
$$
p_{N,k}(\phi)=\sup_{x,\,|\alpha|\leq k}(1+|x|^2)^{N/2}|D^\alpha\phi(x)|
$$
is finite. Convergence means convergence in every such seminorm. A <tempered distribution> is a continuous linear functional on this Fréchet space; $u_j\to u$ in $\mathcal S'$ means $u_j[\phi]\to u[\phi]$ for every test function $\phi$.
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